The String Coffee Table

Effective theories and gravity

Posted by Robert H.

To rebut fears that having my own blog would let me forget about the coffee table i would like to share some thoughts that I would like to here your comments on. The theme is effective theories and how they fit together with gravity.

As long as we are not entirely sure about the nature of the final theory (I know, string theory is the final theory but let us forget about this for a second) we have to keep an open mind about the fact that all theories that we deal with are only effective theories and that they are obtained by some coarse graining procedure from more fundamental theory. Then the usual renormalization group reasoning tells us that our low energy measurements should be independant of irrelevant deformations of this procedure. Especially, as long as our low energy observational scale is well below the coarse graining scale, low energy physics should be invariant under moving that scale by means of integrating out or in high energy degrees of freedom.

To be a bit more specific about the kinds of models I have in mind let me tell you about a U Chicago exam problem that Gavin Polhemus (the coauthor on my first string theory paper) once told me about:

Find the energies of the lowest energy modes of a particle in a 2d box that has the shape of a highly excentric ellipse!

Say, the particle is confined to $x^{2} + y^{2}/e^{2} \lt 1$ for small $e\ll 1$. Then the idea is to realize that the bounces in the $y$ direction are much faster than those in the $x$ direction and are thus at a higher energy scale and can be integrated out.

By that I mean we use the Born Oppenheimer approximation and treat $x$ as a classical constant variable while solving the Schr"odinger equation for $y$. This is the ordinary particle in the box of size $2e\sqrt{1-x^{2}}$ and this has a zero point energy of

The trick is now to interpret $E_0(x)$ as an effective potential for a one dimensional problem envolving only $x$.

Note one feature of this integrating out procedure: In the original 2d problem, no matter where in the ellipse the particle is, there is no potential energy, all there is is kinetic energy. However, in the effective 1d theory, suddenly what used to be the kinetic energy of the motion in the $y$-direction is now interpreted as potential energy of $x$. This in fact is quite generic of effective theories: Effective potentials often come from kinetic energy in the microscopic theory.

Now you probably wonder why I care, in the end it should not matter what type of energy we have. Well, this is only true as long as we are not dealing with gravity. Gravity appearantly can diffentiate between kinetic and potential energy. One way to see this is to consider the typical cosmology setting of a FRW metric coupled to a scalar field. This problem is again one dimensional as everything (the scale factor and the scalar) only depend on time.

Usually, the Friedmann equations are expressed in terms of the energy density and pressure of a perfect fluid:

and we have to specify $\rho$ and $P$ in terms of the properties of the scalar field. It turns out that $\rho = T + V$ and $P=T-V$ where $T$ is the kinetic and $V$ is the potential energy of the scalar field. Thus $P$ and therefore the evolution of $a$ depend on how the energy is distributed between kinetic and potential energy.

Now imagine that we coupled the above QM system to gravity: We consider two scalar fields $X$ and $Y$ that are free except that they are confined to $X^2 + Y^2/e^2\lt 1$. Then I would expect one evolution corresponding to pure kinetic energy. Alternatively I would integrate out $Y$ and obtain a different cosmology with partly kinetic and partly potential energy. In terms of an equation of state this the first one would be $w=P/\rho = 1$ whereas the second would be roughly $w=0$.

Please find the flaw in this argument, otherwise I would conclude that dealing with effective theories is not save in a gravitational setting!!!

There are two related issues: First of all, strictly speaking, the effective potential is only defined up to an additive constant. But we know we are not supposed to fiddle with that as that would be a cosmological constant and gravity definitely depends on that.

The other issue is less related but still makes me a bit uneasy: There are all kinds of gravitational entropy bounds on the marked that all roughly say that the entropy of an object is bounded by $S=A/4$, where $A$ is the horizon area the black hole you could turn this object into. The reason I am uneasy with these bounds is that in thermodynamics we usually deal only with differences of entropy and only in a quantum treatment you adopt the convention of setting the entropy of the ground state to 0. This corresponds to a normalization of the partition function and of course only holds if the ground state is non-degenerate.

However, what you call the ground state depends on you level of course graining! For example, in the past I worked on protein folding. There the typical question is, given some sequence of amino acids, what is the ground state? We often joked that the ground state of a protein is some $H_2O$ and $CO_2$, but that was wrong: You can still gain some energy by nuclear processes that turn the protein into a lump of iron.

Obviously, what this joke plays with is that bio-physicists use a different coarse graining thant chemists (they assume molecular bounds are inert) that use a different coarse graining than nuclear physicists (as they treat chemical elements as inert). The upshot is that what looks like a ground state in one coarse graining might actually be composed of many micro states in a more microscopic theory. Thus the biophysicist would assign entropy 0 to the protein in the ground state whereas the chemist would assigne a positive entropy to it and the nuclear physicist would assign an even bigger entropy to it. A string theorist might even find more stringy degrees of freedom and would assign an even bigger entroby to it. And hey, maybe also string theory is only an effective theory and there is an even more microscopic theory behind it that would know about even more microstates!

Well, thanks to Beckenstein, we don’t have to worry about this, we can meassure it: We just throw the protein into a black hole and measure how the horizon area increases. This you can translate into an upper bound of the entropy of the protein and this in turn tells you that you cannot infinitely find more fundamental theories with more and more degrees of freedom. Thus you can for example measure if there is room for a more fundamental theory than string theory! Isn’t that great? And all this only envolves low energy physics: You just have to slowly lower a protein into a black hole.

You think I am joking? Well, have a look at papers like this. Here, Andreas derives bounds on the quotient of viscosity and entropy density from holographic arguments. And he even finds that actual fluids (as found in the CRC) are only one order of magnitude away from the bound! But this should make you wonder how the absolute entropy in those tables is defined. And indeed, the tables assume that the chemical composition is kept fixed when the entropy is extrapolated to zero temperature. But we just found that from a nuclear perspective the entropy should be bigger because we have to count nuclear degrees of freedom as well. Hmm.

Please help, I’m worried.